Statistical Mechanics in the Canonical Ensemble Outline of the Formalism and Some Examples

The Canonical Ensemble: Outline of General Formalism The Probability that the system is in quantum state r at temperature T is: Ur  εr = energy of state r. Minus in exponentials! “Partition Function” The Canonical Distribution gives “The probability that a system in contact with a heat bath at temperature T should be in state r”.

 Only states with the lowest Ur  εr = energy of state r. - “Partition Function” - pr  Probability that a system in contact with a heat bath at temperature T is in state r. At Low Temperatures:  Only states with the lowest energies εr (Ur ) have a reasonable chance of being occupied.

Ur  εr = energy of state r. - “Partition Function” - pr  Probability that a system in contact with a heat bath at temperature T is in state r. At High Temperatures:  States with higher lying energies εr become more & more likely to be occupied. Clearly, all microstates are not equally likely to be populated.

Canonical Distribution Often, there are huge numbers of microstates r that can all have the same energy. This is called DEGENERACY & is indicated by a degeneracy factor g(Ur). In this case, the summations are over each individual energy level rather than sum over each microstate. The sum is over each different energy Ur. The degeneracy factor g(Ur) is the number of states with energy Ur. The probability p(Ur) is that of finding the system with energy Ur.

Entropy in the Canonical Ensemble The system of interest A is in equilibrium with a heat bath A' for which the energy fluctuates & the probability of finding it in any particular microstate is variable. Goal: Calculate The Entropy S for system A. From S, all thermodynamic variables can be calculated. System A is in contact with a heat bath of (M-1) subsystems of the one of interest. Each subsystem may be in one of many microstates. ni  number of subsystems in the ith microstate. As we’ve seen before, the number of ways of arranging n1 systems of microstate 1, n2 systems of microstate 2, n3….is: A ' A

Other, equivalent forms for Entropy are: After some manipulation, it can be shown that, in The Canonical Ensemble, The Entropy can be written: This is a General Definition of Entropy & holds even if the probabilities of each individual microstate are different. After more manipulation, it can be shown that Other, equivalent forms for Entropy are:

Entropy in The Canonical Ensemble This general Definition of Entropy, in combination with The Canonical Distribution allows the calculation of all of the system thermodynamic properties:

Helmholtz Free Energy Internal Energy: Note: This means that Z can be written: Z  exp[-F/(kT)] Internal Energy: Ū  Ē  Average Energy of the system Helmholtz Free Energy, F. F = Ū - TS  Average Helmholtz Free Energy The Partition Function Z is much more than a normalizing factor for probability. Z acts as a “Bridge” linking microscopic physics (quantum states) to the energy & so to all macroscopic properties of a system.

Helmholtz Free Energy  F is a state function. Now, we’ll calculate some other thermodynamic properties of the system. Ignore the fact that Ū is an average & let U = Ū. Use definitions for thermodynamic variables. For an infinitesimal, quasistatic reversible change: 

Using the properties of partial derivatives gives: Equation of State Entropy Using The Canonical Ensemble, the energies of the microstates of the system are directly linked to macroscopic thermodynamic variables such as pressure & entropy.

Example: Heat Capacity at constant volume: Measurable Thermodynamic Variables: 2nd Derivatives of Helmholtz Free Energy F Various Elastic Moduli are stress/strain or force/area divided by a fractional deformation: Example: Bulk Modulus K Various Thermodynamic Properties Example: Heat Capacity at constant volume:

How large are the fluctuations? Mean Internal Energy Ū  Thermal Average of the system Internal Energy. The actual internal energy fluctuates due to the system interacting with the heat bath. How large are the fluctuations? Are they important?

Fluctuations in Internal Energy A measure of the departure from the mean is the standard deviation, as in any statistical theory. Some detailed manipulation shows that

N  Number of Particles in the system. The Variance The relative fluctuation in energy (U/Ū) gives the most useful information. More manipulation shows that: Ū & CV are extensive properties proportional to the size of the system. So, they are both proportional to N  Number of Particles in the system. This implies that

For Macroscopic Systems with ~1024 particles, the relative fluctuations (U/Ū) ~ 10-12 The fluctuations about Ū are very tiny, which means that U & Ū can be considered identical for practical purposes. Based on this, it is clear that Macroscopic Systems interacting with a heat bath effectively have their energy determined by that interaction. Similar relationships also hold for relative fluctuations of other macroscopic properties.

Summary: The Canonical Ensemble pi  Probability that the system at temperature T is in the state i with energy Ui. Partition Function  Sum over All Microstates: Mean Energy: Helmholtz Free Energy:

The Canonical Ensemble is consistent with Boltzmann’s Definition of Entropy: Boltzmann’s Definition of Temperature: Boltzmann’s Definition of Pressure (Equation of State):